Noether's Theorem

Noether's Theorem applies to any problem which can be written in terms of a variational principle. In the case of a single independent variable the resulting Euler-Lagrange Equation is an ordinary differential equation (or system of equations). The order of the equation need not be second. The theorem applies equally well to many independent variables. The essence of the theorem is to calculate the infinitesimal transformations under which the Action Integral is invariant. To each of these transformations there corresponds a symmetry and from that symmetry a first integral is readily calculated. We concentrate on problems arising in classical mechanics, but must emphasise that the applications have a much wider scope.